Given Finite Nature, what we have at the bottom is a Cellular Automaton of some kind.
-- Edward Fredkin.

The idea that the universe we perceive is the projection of a running computer simulation is still
novel, but it is being considered in some quarters. Chief among the proponents is Edward Fredkin
who, in 1992, published two papers giving the matter serious thought.[1]

Fredkin has been associated with both computer science and physics throughout his career, and it
was perhaps his familiarity with both that led him first to suspect, then to believe that the universe
is a cosmic computer program. In the early 1950s, Fredkin was one of the original computer
hackers on one of the original computers. He later helped to establish the prominent research
facility for computer theory and applications at MIT. He also studied with some of the greatest
physicists of our time, including Richard Feynman. In his work, Fredkin made fundamental
contributions to both physics and computer science, frequently driven by his unwillingness to
accept conventional wisdom in either field.

Fredkin begins his exploration of the universe with a simple observation: from the time Max
Planck coined the terms "quantum" and "quantized," it has happened that every physical
phenomenon susceptible of precise measurement has turned out to be "quantized." Einstein
proposed in 1905 that light was so quantized that it was sometimes best thought of as "chunks" of
energy, i.e., particles dubbed photons; Neils Bohr proposed in 19xx that electrons associated with
an atomic nucleus were "quantized" in their ability to fill "shells" (see Chapter xx); the hypothetical intrinsic motion of quantum particles -- angular momentum or "spin" -- is an either-or quality with no in-between. We may go back further, to the theory of atoms which showed that everything -- earth, air, fire, water -- is comprised of discrete chunks of atoms and molecules as grainy as the sand, regardless of how smooth and continuous they might appear.

While other phenomena -- notably position and momentum, which is to say space and time --
remain undetermined, Fredkin reasoned that the clear tide of scientific progress seems to favor
discreteness over continuity. As we discussed earlier (Chap. xx), what might seem continuous on
the order of a motion picture might turn out on closer inspection to be a series of discontinuous
events like frames of still pictures flashing by faster than we can count them. Fredkin speculated
that it is most likely that all phenomena eventually will be found to be step-wise, discontinuous,
quantized in their nature, rather than smooth and continuous. That is to say that nothing is
continuous, despite appearances created by our limited powers of observation. Not light, not
"fields," not space, not time.

Fredkin saw that a necessary corollary of all things being discrete and stepwise is that all things are
thereby made finite. That is, there will be no infinity and no infinitessimal in our physics because
those bothersome concepts occur only in the gaps between measurements, of which there are none
if one discrete measurement abuts the next.

Fredkin called his thesis "Finite Nature," and he saw that this hypothesis, if true, poses the greatest
challenge for physical theory. It is nothing less than Zeno's paradox in the flesh: if all things are
discrete -- including most fundamentally space and time -- then nothing can change. There can be
no evolution from the state that exists at one moment to the state that exists at the next moment
because there is no transition, no morphing, no sliding, no movement, nothing in between. The
thing exists at one moment, then it exists at the next moment. How, then, to account for the
observed fact that the thing is different from one moment to the next?

The way to account for change is to see "properties" as finite sets of information. In dealing with a
set of information, we see that it can change and transform only according to rules laid down by
another set of information. This is the "process" of changing information, and it is the very
definition of "information processing." It is also the very definition of a digital computer. Let us
examine this two-part concept -- "digital" and "computer" -- according to Fredkin's analysis in
Finite Nature.

Digital. The hypothesis that all things are discrete implies first that all things are finite. That is,
there is no infinity and there is no infinitesimal. ("Whew!" says Zeno. "Whew!" says the
quantum physicist.) No matter how big the numbers get, and no matter how small, there will be an
end to them, and it will be a definite number. That's the difference between something that is
infinite (or infinitesimal) and just really-really-big (or really-really-small). There is an end. It is
definite. It is finite. Achilles will come to the point where there are no more half-ways, and so he
will cross to the finish line. The quantum physicist will reach the end of the calculation, with
nothing between him and zero.

A property or quality which is discrete and finite can be described exactly. For example, if a coin
has the discrete and finite property of being heads or tails, then it cannot be anything in
between.[2] Accordingly, when we observe that a coin is "heads," we describe the coin's state
of-flip exactly.

Where the possible states are so limited, there is an exactly limited amount of description that can
be given before we run out of things to describe, and ways to describe them. The state-of-flip
(heads or tails, or simply "h" or "t") may be one property; and the place-of-minting (Philadelphia,
Denver or San Francisco, or simply "P," "D" or "S") another; and the denomination (nickel, dime,
quarter, etc.) may be another; and so on. Each of these properties, being limited to a certain
number of possibilities, can be described with a single choice from among its finite
possibilities.[3] We can see that with these restrictions, each coin can be described exactly by
stating the value of each property of the coin.

Given Finite Nature, there are no approximations, no subjective values. A collection of three
coins, in our example, can be arranged and rearranged in only so many ways before one runs out
of possible combinations. Flipping one or another will yield different state-of-flip arrangements
with exactly 8 possibilities, no more and no fewer:[4]

hhh, hht
hth, htt
thh, tht
tth, ttt

Similarly, the possible place-of-minting values for these three coins will be exactly 27, no more
and no fewer. Choosing to describe both of these properties for our collection of three coins (Ph-Ph-Ph, Ph-Ph-Pt, etc.) increases the number of possibilities in a tidy mathematical way, so that
resulting number of possibilities, while large, remains exact and decidedly finite.[5]

If all of space-time and all physical processes are fundamentally discrete, then for any given unit of
space-time (i.e., for any volume of anything, or even a volume of nothing) there will be a limited,
finite number of combinations that will describe all of the possible states of everything contained
within that volume. Put another way, for every volume of space-time there will be a definite and
finite amount of information contained within it.

What is more, a property or quality which is discrete and finite, and which therefore can be
described exactly, can be described exactly by a number. How fast? Sixty-three m.p.h. (as
opposed to the alternative non-finite description, "rapidly"). The data itself, and thus the number
itself, is the relevant information. In our coin example, heads could be represented by "0"
and tails by "1", so that heads-heads-tails (or hht) could be represented by "0-0-1" or just "001."
There is no difference in the information content between "heads-heads-tails" and "hht" and "001."
What is more, there is no difference in the information content between the statement "three coins are lying on a
table; two coins are heads up and one coin is tails up," and the symbolic number "001." So long as this property is discrete and finite,
the number is all we need for a complete and exact description.

The interpretation of these numbers is easily accomplished simply by keeping track of what each
digit is supposed to represent. In the previous example, the numbers represented a coin's state of
flip. We could as easily have a secret code: if I say "one," you will know that I mean "by land"; if I
say "two," you will know that I mean "by sea." Thus, by saying "one," I convey to you the entire
sense that the-British-invasion-force-has-been-spotted-and-it-is-traveling-overland-to-attack-our
militia-positions. And you have your further instructions that I intend by this signal for you to ride
off on your horse to alert our militia forces to the nature of the threat. Because the interpretation of
the numbers is necessarily arbitrary,[6] we will leave aside the question of interpretation for the
moment to focus on the numbers themselves.

This is digital: numbers, digits. Information expressible as numbers. Digital information is a
wonderful thing.

Computer. Now that we have all of these numbers exactly describing every property of every
thing in existence, what shall we do with them? We can print them out in a very large spreadsheet,
or as a very long string of numbers. That will give us a complete description of the universe at one
instant -- every last pixel of one frame of the motion picture -- but it will hardly do as an arena in
which to live out our lives. How do we get to the next frame, preferably without starting from
scratch? From one moment to the next -- from one time-step to the next -- the information must be
transformed completely to a new and different, but equally finite, set of information describing
every property of every thing in existence as it exists at that next time-step.

We have one complete set of information, and the next complete set of information should bear
some relationship to what we have. Therefore, we would like to change the present set of
information into a new set of information that is somehow related. To do so, we will need a
separate body of information consisting of rules for how this change should take place.

Put in terms of numbers, we have one complete set of numbers, and the next complete set of
numbers should bear some relationship to the numbers we have. Fortunately, this is quite easy to
do with numbers. We simply refer to a rule, or formula, or algorithm, or equation, by which we
can transform one number into another. Here's a simple rule: multiply each number by two. The
next set of numbers, according to this rule, will each be double the value of the previous number.
If we perform this operation on each number in our spreadsheet, we will have a new spreadsheet
with new numbers (each of which is twice the value of the corresponding number of the old
spreadsheet), representing the next frame in the progression of our movie.

Taking one set of information represented by numbers, applying one or more rules for changing
the numbers, and thereby obtaining a new set of information represented by a new set of numbers,
is the process we call computing. For obvious reasons, it is also called information processing --
applying a process of transformation to a set of information. The information is represented by the
numbers, stored in the computer; and the process of transformation is supplied by the rule or rules,
stored elsewhere in the computer (also as numbers).

The clock -- a computer's quantized time. The computer's information is represented
symbolically by the arrangement of binary switches in the computer's memory. The "state" of the
computer is the aggregate arrangement of these memory bits. Consequently, when we consider the
state of the computer, we must look at the arrangement in static, fixed form as it exists in its initial
state and at the end of a step of programming. The arrangement cannot be changing as we consider
it, because the "meaning" represented by the state of the computer depends on the relationship of
each bit to every other bit.

The computer considers the information coded in its "state" and applies its programming rules to
that information, changing the arrangement of the memory bits according to the rules. As the
memory bits are being changed, the internal arrangement of the computer is in a state of transition.
If the process were stopped in the middle of this transitional phase (before all of the rules for this
"step" were fully carried out), an observer looking at the arrangement would not be able to extract
any meaning at all. The overall arrangement would be "wrong" because the application of the rules
had not been completed. This would be a computer crash.

To illustrate, suppose the memory bits were soldiers lined up in a row, and the rule was "take one
step forward; then take another step forward; then take one step back." In our computer analogy,
the sergeant must give these orders, one at a time, to each individual soldier, and each soldier then
must carry out the orders. If we halted the process mid-application, we might see a very ragged
line of soldiers because some had taken two steps forward and one step back, some had only taken
two steps forward, and some had not yet moved. On the other hand, if we allowed the rule to be
applied fully, we would see a tidy row of soldiers which had neatly advanced one step. If you had
closed your eyes before the sergeant started (very quietly) giving orders, and opened them after he
was finished, you might imagine that all of the soldiers had advanced one step all together. You
would be wrong, but the net effect would be the same.

A computer achieves this tidy progression through a hardware technique called single clock. As
Fredkin explains, "Single clock means that when the computer is in a state, a single clock pulse
initiates the transition to the next state. After the clock pulse happens, various changes propagate
through the computer until things settle down again. Just after each clock pulse the machine lives
for a short time in the messy real world of electrical transients and noise. At the points in time just
before each clock pulse the computer corresponds exactly to the workings of a theoretical
automaton or Finite State Machine."

The single clock synchronizes the information processing in a computer, so that the programming
can be applied in steps. At the tick of the computer's "clock," the programming is applied and the
arrangement of the memory bits begins to change. After all memory bits have been affected, the
clock tick is finished and the programming rule has been fully applied.

The computer's process of transition, unknown to Zeno, is not reflected by the state of the
computer's memory itself, but exists in the abstract application of the rule or rules. We do not
"see" the transition, nor even the process of transition, because there is not and cannot be any
provision for "reading" the data while the transition is in progress. As it happens, this is exactly
the case with quantum mechanical transitions, or "quantum leaps," which were anticipated by Zeno
(in the stadium paradox, see Chapter xx) but dismissed as absurd and impossible by Aristotle.[7]

Edward Fredkin was in a unique position to discover the Rosetta Stone that would connect the
seemingly disparate languages of physics and computer science. The link turned out to be cellular
automata -- a method of programming computers according to a small number of simple rules
which, when run repeatedly over a large number of cycles, developed the same dense complexity
we observe in the physical systems of the natural world. Cellular automata programs have been
developed to mimic the behavior of gas volumes, electrons traveling down a copper wire, ant
colonies, and most famously the course of biological evolution in the "Game of Life." Fredkin saw
applications of the cellular automata computer architecture everywhere he looked in physics. He
began to believe that the match couldn't be a mere coincidence, and he formed the idea which has
come to be known as the "Fredkin Hypothesis": the universe is a computer programmed according
to cellular automata principles. More precisely, the universe is the manifestation of the calculations
of a computer programmed according to cellular automata principles.

The Cellular Automata computer architecture. The cellular automata computer type of
architecture provides a full analogy for the concept of discrete blocks of space-time which evolve,
both individually and collectively, from moment to moment. This architecture denominates each
computing unit as a "cell," which operates as though it were an independent computer with a
simple set of programming instructions (although it is more accurately one of a large number of
identical subroutines). Because the cell is operating independently and automatically, in robot-like
fashion according to its rules, the cell is known as an "automaton" as though it were a robot
performing its tasks mindlessly. The aggregation of many of these computing cells comprises the
"Cellular Automata" computer architecture.

There is no difference in principle between the individual cellular automaton and any other
computer. Both consist of a block of memory which is acted upon by a set of programming
instructions. The difference in practical terms is that the cellular automaton operates according to a
severely restricted set of instructions (programming), and so requires a comparatively modest
amount of memory. With these limitations, we can create a vast number of cellular automata using
our finite memory and programming resources. The key to the utility of the cellular automata
("CA") computer architecture is that when we assemble this vast number of simple, independent
computing units, they can interact among themselves in breathtakingly complicated ways.

To illustrate, let us consider two simple cellular automata, side by side, whose only function in life
is to have a color -- either blue or green or red. Each cell has only one rule to follow: it should
ponder the color of the automaton next to it, and take the color which is next in line alphabetically.
Thus, if its neighbor is green, it should take the color red; if red, it should take the color blue; if
blue, it should take the color green. We can then watch as the situation changes over time (having
arbitrarily assigned the colors Blue and Green to the automata to begin with):

Time step

Cell #1Cell #2

Application of the rule

1:

Blue's neighbor is green, so it will become red; green's neighbor is
blue, so it will become (remain) green.

2:

Red's neighbor is green, so it will become (remain) red; green's
neighbor is red, so it will become blue.

3:

etc.

4:

etc.

5:

etc.

6:

etc.

. . . and so on. We have a pattern which, in this case, will repeat itself every six steps. Exactly
the same rule is applied at each time step, but the pattern is slightly complex.

It is the interaction among the cellular automata which causes their states to evolve, and to evolve in
a way that bears a relationship between past, present and future such that a pattern emerges.

Cellular Automata theory considers each automaton as a "cell" surrounded by other cells. Each cell acts independently in the sense that it follows its own rule or rules when deciding what it
shall become in the next instant of time (just as in our simple example above). The rule of each cell
is exactly the same as that of every other cell, and each takes into account the state of each of its
neighbors and determines its next future state on the basis of the input received from its neighbors'
present state, according to the rule.

The simplicity of CA architecture gives rise to a vast complexity of interaction which, in turn,
produces pleasing patterns of apparent order as the scale increases. An examination of CA
architecture shows that it has the potential to model all varieties of interactions in the natural world
for, like a cellular automaton, a quantum unit in our world interacts with its neighbors to produce
change according to strict mathematical rules which take into account the quantum unit's own
discrete, individual state and the impinging states of its neighbors.[8]

Fredkin concludes: "Given Finite Nature, what we have at the bottom is a Cellular Automaton of
some kind." And, because "Automata, Cellular Automata and Finite State Machines are all forms
of computers," this is to say that at the bottom of the physics of the natural world, we have a
computer of some kind.

Fredkin's conclusion that the particular kind of computer we have at the bottom is a cellular
automata computer requires some additional understandings not fully discussed in his early papers.
Nevertheless, we can see how the thesis that all things are finite introduces a compelling logic.
Accepting Zeno's proof that a physical finite system cannot evolve, the premise leads to the
conclusion that all things are computer:

Finite systems can be exactly described with finite information.

Where a system can be exactly described, the information which describes it is fully

equivalent to the system itself.

A finite set of information cannot transform into anything other than another finite set of

information.

A finite set of information cannot transform into another finite set of information except by

some process of information transformation.

Finite information can be fully represented by symbols such as numbers.

Finite, symbolic information such as numbers can be transformed by the application of

rules to obtain other numbers.

The application of rules to transform numbers into other numbers is computation.

As of this writing, Fredkin's ideas continue to meet with resistance from the moment he states his
premise -- the very concept striking some as on par with alien abductions or some other cultic
doctrine. Nevertheless, as physics moves closer to a paradigm of information, and farther from
the mental image of specks and particles, there is some interest in what might be the implications of
a universe operating on principles of information exchange.

In his book The Bit and the Pendulum, science writer Tom Siegfried surveys contemporary
scientific thought from a number of fields and observes that scientists increasingly are resorting to
information theory and the metaphor of information processing (computing) to explain a wide
variety of problems in their fields. He includes an obligatory reference to Edward Fredkin's view,
assessing it as follows:

In a column I wrote in 1993, I suggested that the chances that [Fredkin] is right are
about the same as the chances that the Buffalo Bills will ever win the Super Bowl.
So far, so good. The Bills have never won [as of 2000]. But they have made
some of the games interesting.[9]

Assuming that Buffalo maintains its NFL franchise, the chances are that, eventually, it will win a
Super Bowl championship, though perhaps not in our lifetime. This is why I like Siegfried's
statement of the odds -- it lends an air of statistical inevitability to the triumph of the Fredkin
Hypothesis that matches the inexorable march of quantum mechanics toward the goal line of pure
mathematics. Fredkin has shown that the human mind can imagine what underlies the physics of
our world, and without too much difficulty at that. In the next chapter, we will step back from the
full vision of a computer-generated universe to consider the lowly bit of information and its relation
to the quantum.

The difference between these finite coins and the change in your pocket is that the coins in
our illustration cannot be rolling, or on edge, of anything other than heads or tails. That is what
we mean by a discrete, finite property.

In this scheme, even such seemingly continuous properties as the general condition of the
coin are limited to a set of choices -- such as "proof," "mint," "uncirculated," and "circulated" --
rather than being described along a full and infinite range of wear-and-tear. Although this type
of limitation goes against common sense, it is required by Finite Nature and, in fact, it is a good
illustration of the baffling step-wise behaviors of quantum mechanics.

The reader can see why this limitation applies only to discrete properties. If we were to allow
intermediate states such as the coin standing on its edge, and all of the instantaneous
positions through which it can pass while rotating as it is being flipped, then there would be an
infinite number of possibilities for the coin's state-of-flip property. The difference between
eight possible states and an infinite number of possible states is our hypothesis that the coin
can only exist in a complete state of being "heads," or alternatively in a complete state of being
"tails," but never in between. Oddly, the scientific proofs that natural phenomena are
fundamentally discrete rely on just this type of puzzling limitation.

The meaning is anything the programmer wants it to be. As David Eck puts it, "Suppose I were
to point to some particular sequence of bits inside a computer and ask what it represents.
Without further information, the answer could be almost anything--the current date, the color
of some particular pixel on the screen, the board position in a game of computer chess, Joe
DiMaggio's batting average in 1939 .. .. What it actually means is determined not just by the
sequence of bits but also by the physical structure of the computer itself, by the overall
structure of the data encoded in the computer, by the program that is running, and by the
intentions of the person using the computer." D. Eck, The Most Complex Machine, at 11(A.K.
Peters, Ltd., Wellesley, MA 1995).

In his paper "Finite Nature," Fredkin does not address the ramifications of the E-P-R effect,
whereby a quantum unit can be "influenced" by other quantum units far removed from its
immediate neighborhood. However, other aspects of computer architecture may provide the
most plausible explanation for this phenomenon. See R. Rhodes, "A Cybernetic
Interpretation of Quantum Mechanics" at 13 (1999).